Hello, 52090!

If we *must* use Calculus, the approach and set-up is quite ugly.

A water line runs east-west. A town wants to connect two new housing developments

to the line by running lines from a single point on the existing line to the developments.

One is 3 miles north of the existing line; the other is 4 miles north of the existing line

and 5 miles east of the first development.

Find the place on the existing line, relative to the two developments,

to make the connection that minimizes the total length of the new line. Code:

* B
* |
A * * |
| * * | 4
3 | * * |
| * * |
* - - - * - - - - - *
C x P 5-x D

Development is 3 miles from the line: .

Development is 4 miles from the line: .

Let be a point on .

Let , then

We want to minimize the distance:

In right triangle

In right triangle

The total distance is: .

. . and that is the function we must minimize.

I'll wait in the car . . .

.